Key concepts we covered

• Population vs. sample
  • Population: The complete set of all items we’re studying
  • Sample: A subset of the population we actually measure
• Parameters (population) and statistics (sample)
  • Central tendency: how data clusters around a middle value
    • mean (average), median (middle value), mode (most common)
  • Spread/dispersion: how data points vary from each other
    • variance (average squared deviation), standard deviation (square root of variance)
• Confidence intervals – we’ll explore this further today!

Overview

We will focus on the fundamentals behind observational studies this week.

Observational studies: two common types

Sampling designs

What is “random” sampling?

We know that…

From the previous lecture

• Sample mean is a good measure of central tendency (the “middle” of our data).
• Sample variance is a good measure of dispersion (how spread out our data is).
• Sample size affects the precision of the sample mean (more samples = more precise).

Combining an estimate with its precision

A confidence interval (CI) is:

• A range of values that likely contains the true population value
• Like saying “We’re 95% confident that out of 100 samples, 95 of them will be within this range”
• Crucial for hypothesis testing and estimation, the basis of statistical inference

You will often see CIs in scientific papers, reports, and news articles.

Calculating confidence intervals

What we need

1. Estimate of the population parameter, i.e. the sample mean (\bar{x}) - our best guess of the true value

2. Critical value (t_{n-1}) - a number that represents our chosen confidence level (like 95%)

3. Standard error of the estimate, SE of the mean (SE_{\bar{x}}) - tells us how precise our mean is

All these (mean, standard error and critical value) are combined to form the confidence interval.

Breakdown

In general, a CI has the form: \text{estimate} \pm \text{margin of error}

where the margin of error is a function of the standard error of the estimate:

\text{estimate} \pm (\text{critical value} \times \text{standard error (estimate)})

where the critical value is based on the sampling distribution of the estimate i.e. the t-distribution.

Formula for 95% Confidence Interval (CI)

\bar{x} \pm \left(t_{n-1} \times \frac{s}{\sqrt{n}}\right)

Step-by-step calculation by hand

1. Calculate the sample mean, \bar{x} (add all values and divide by number of samples)

2. Calculate the sample standard deviation, s (measure of spread in your data)

3. Determine the standard error of the mean, SE_{\bar{x}} = \frac{s}{\sqrt{n}} (how precise your mean estimate is)

4. Look up the t-value, t_{n-1}, from the t-distribution table for the 95% confidence level and n-1 degrees of freedom.

   • This is a specific number based on how many samples you have

5. Compute the margin of error: \text{Margin of Error} = t_{n-1} \times SE_{\bar{x}}

6. Finally, the 95% CI is: \bar{x} \pm (t_{n-1} \times SE_{\bar{x}})

You need to be able to calculate this by hand/calculator.

More definitions

Degrees of freedom (df)

The number of values in a sample that are free to vary while still maintaining the same statistic.

\text{df} = n - 1

where n is the number of samples.

More definitions

t-critical value

A number that helps determine how wide to make your confidence interval.

• Based on your confidence level (e.g. 95%) and sample size
• Larger t-values = wider intervals = more confidence but less precision
• Smaller sample sizes = larger t-values (because we’re less certain)

More definitions

t-distribution

A probability distribution that accounts for the uncertainty when estimating from small samples.

• Similar to the normal “bell curve” distribution, but with heavier tails.
• With few samples (small n), the t-distribution is wider, reflecting greater uncertainty.
• As sample size increases, the t-distribution gets closer to the normal distribution.

library(ggplot2)
library(gganimate)

# Variable to control animation speed (higher values = slower transitions)
anim_speed <- 1

# Create a sequence of x values for the plot
x <- seq(-4, 4, length.out = 400)

# Degrees of freedom: 1 through 5 then even numbers from 6 to 30
dfs <- c(1:5, seq(6, 30, by = 2))

# Prepare data for t-distribution curves with a new "df" column
t_curves <- do.call(rbind, lapply(dfs, function(df) {
  data.frame(
    x = x,
    density = dt(x, df),
    df = df
  )
}))

# Create data for the standard normal distribution (static)
normal_curve <- data.frame(
  x = x,
  density = dnorm(x)
)

# Plot with gganimate: animate t-distribution curves (each frame shows one df)
p <- ggplot() +
  # Static normal distribution curve
  geom_line(
    data = normal_curve, aes(x = x, y = density),
    color = "black", linetype = "dashed", size = 1
  ) +
  # t-distribution curve that will animate
  geom_line(
    data = t_curves, aes(x = x, y = density, color = factor(df)),
    size = 1
  ) +
  labs(
    title = "Degrees of Freedom: {closest_state}",
    x = "x",
    y = "Density",
    subtitle = "Dashed line = Normal distribution; Solid line = t-distribution"
  ) +
  theme(legend.position = "none") +
  transition_states(states = df, transition_length = anim_speed, state_length = anim_speed)

p

Interpreting confidence intervals

• Confidence intervals depend on a specified confidence level (e.g. 95%, 99%) with higher confidence levels producing wider intervals (i.e. more conservative).
• Another way to think of it: a range of values that we are fairly sure contains the true value of the population parameter.

Soil carbon

Soil carbon content was measured at 7 locations across the area. The amount at each location was 48, 56, 90, 78, 86, 71, 42 tonnes per hectare (t/ha).

soil <- c(48, 56, 90, 78, 86, 71, 42)
soil

[1] 48 56 90 78 86 71 42

What is the mean soil carbon content and how confident are we in this estimate? How this is calculated depends on whether we used simple random sampling or stratified random sampling.

Mean and 95% CI

Step-by-step calculation

Implementation in R

Manual calculation

# Step 1: Calculate the sample mean of soil carbon content
mean_soil <- mean(soil)

# Step 2: Calculate the sample standard deviation of soil carbon content
sd_soil <- sd(soil)

# Step 3: Calculate the standard error of the mean (SE)
se_soil <- sd_soil / sqrt(length(soil))

# Step 4: Calculate the t-critical value for a 95% confidence interval
t_crit <- qt(0.975, df = length(soil) - 1)

# Step 5: Calculate the margin of error (t_crit * SE)
# and then determine the lower and upper bounds of the confidence interval
ci <- mean_soil + c(-1, 1) * (t_crit * se_soil)

# Step 6: View the calculated 95% confidence interval
ci

[1] 49.84627 84.72516

There are ways to calculate this in R quickly, but it is important to understand the manual calculation.

Questions

• How precise is our estimate?
  • Our margin of error is about 17.43 t/ha, which is roughly 26% of our mean value.
• How big a change must there be to estimate a statistically significant change?
  • Changes larger than about 17.43 t/ha would likely be statistically significant.
• Can we sample more efficiently?
  • Yes! This is where stratified sampling becomes valuable.

To answer these questions, we need to compare simple random sampling with a hypothetical stratified random sampling design (i.e. what if we had considered stratification before sampling?)